| L(s) = 1 | − 1.62·3-s − 5-s + 4.42·7-s − 0.358·9-s − 1.81·11-s − 1.06·13-s + 1.62·15-s − 1.23·17-s + 4.12·19-s − 7.18·21-s − 4.61·23-s + 25-s + 5.45·27-s + 0.847·29-s − 0.697·31-s + 2.95·33-s − 4.42·35-s + 3.45·37-s + 1.72·39-s − 3.20·41-s − 10.5·43-s + 0.358·45-s + 4.77·47-s + 12.5·49-s + 2.00·51-s + 3.05·53-s + 1.81·55-s + ⋯ |
| L(s) = 1 | − 0.938·3-s − 0.447·5-s + 1.67·7-s − 0.119·9-s − 0.548·11-s − 0.295·13-s + 0.419·15-s − 0.299·17-s + 0.946·19-s − 1.56·21-s − 0.962·23-s + 0.200·25-s + 1.05·27-s + 0.157·29-s − 0.125·31-s + 0.514·33-s − 0.747·35-s + 0.567·37-s + 0.276·39-s − 0.501·41-s − 1.60·43-s + 0.0533·45-s + 0.695·47-s + 1.79·49-s + 0.280·51-s + 0.420·53-s + 0.245·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 + 1.62T + 3T^{2} \) |
| 7 | \( 1 - 4.42T + 7T^{2} \) |
| 11 | \( 1 + 1.81T + 11T^{2} \) |
| 13 | \( 1 + 1.06T + 13T^{2} \) |
| 17 | \( 1 + 1.23T + 17T^{2} \) |
| 19 | \( 1 - 4.12T + 19T^{2} \) |
| 23 | \( 1 + 4.61T + 23T^{2} \) |
| 29 | \( 1 - 0.847T + 29T^{2} \) |
| 31 | \( 1 + 0.697T + 31T^{2} \) |
| 37 | \( 1 - 3.45T + 37T^{2} \) |
| 41 | \( 1 + 3.20T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 - 4.77T + 47T^{2} \) |
| 53 | \( 1 - 3.05T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 - 9.43T + 61T^{2} \) |
| 67 | \( 1 - 4.76T + 67T^{2} \) |
| 71 | \( 1 - 0.533T + 71T^{2} \) |
| 73 | \( 1 + 7.64T + 73T^{2} \) |
| 79 | \( 1 + 9.62T + 79T^{2} \) |
| 83 | \( 1 + 6.01T + 83T^{2} \) |
| 89 | \( 1 + 6.21T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.48674409451086900052931446814, −6.88634946067532080342897789028, −5.87308374073477706880596139041, −5.35849556332201492360488011872, −4.79479984264219487340544391726, −4.20116805809227019825419048355, −3.08013178892788415757231656670, −2.10085883885856545653107520726, −1.13948588617764530255932568053, 0,
1.13948588617764530255932568053, 2.10085883885856545653107520726, 3.08013178892788415757231656670, 4.20116805809227019825419048355, 4.79479984264219487340544391726, 5.35849556332201492360488011872, 5.87308374073477706880596139041, 6.88634946067532080342897789028, 7.48674409451086900052931446814
